There are quite many general results about finitely generated linear groups (e.g. the Malcev theorem that they are residually finite, the Tits alternative for them, etc.). Since at least in the real case these are just groups that act faithfully on $\mathbb{R}^n$ by continuous automorphisms, I wonder whether similar questions have been considered when one replaces $\mathbb{R}^n$ by some slightly more complicated Lie group.
So more precisely, the question is: Let $G$ be a real Lie group and let $\mathrm{Aut}(G)$ be the group of its continuous automorphisms. Let $\Gamma$ be a finitely generated group and suppose there is an injective homomorphism $\Gamma\to \mathrm{Aut}(G)$. Can something general be said about $\Gamma$ for specific $G$ (besides $\mathbb{R}^n$, say when $G$ is the Heisenberg group)?